Antennas and their coupling characteristics for wireless power transfer via magnetic coupling

ABSTRACT

The disclosure provides systems, methods, and apparatus for wireless power transfer. In one aspect, an apparatus configured to receive wireless power from a transmitter is provided. The apparatus includes an inductor having an inductance value. The apparatus further includes a capacitor electrically connected to the inductor and having a capacitance value. The apparatus further includes an optimizing circuit configured to optimize transfer efficiency of power received wirelessly from the transmitter, provided that an amount of the power received wirelessly and provided to a load is greater than or equal to a received power threshold, or optimize the amount of the power received wirelessly from the transmitter, provided that the power transfer efficiency is greater than or equal to an efficiency threshold.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 12/394,033, filed on Feb. 26, 2009 which claims priority benefit from U.S. provisional patent application No. 61/032,061, entitled “Antennas and Their Coupling Characteristics for Wireless Power Transfer via Magnetic Coupling, filed Feb. 27, 2008. Each of the above referenced applications are hereby incorporated herein by reference in their entirety.

BACKGROUND

Our previous applications and provisional applications, including, but not limited to, U.S. patent application Ser. No. 12/018,069, filed Jan. 22, 2008, entitled “Wireless Apparatus and Methods”, the disclosure of which is herewith incorporated by reference, describe wireless transfer of power.

The transmit and receiving antennas are preferably resonant antennas, which are substantially resonant, e.g., within 10% of resonance, 15% of resonance, or 20% of resonance. The antenna is preferably of a small size to allow it to fit into a mobile, handheld device where the available space for the antenna may be limited.

An embodiment describes a high efficiency antenna for the specific characteristics and environment for the power being transmitted and received.

Antenna theory suggests that a highly efficient but small antenna will typically have a narrow band of frequencies over which it will be efficient. The special antenna described herein may be particularly useful for this kind of power transfer.

One embodiment uses an efficient power transfer between two antennas by storing energy in the near field of the transmitting antenna, rather than sending the energy into free space in the form of a travelling electromagnetic wave. This embodiment increases the quality factor (Q) of the antennas. This can reduce radiation resistance (R_(r)) and loss resistance (R_(l))

SUMMARY

The present application describes the way in which the “antennas” or coils interact with one another to couple wirelessly the power therebetween.

BRIEF DESCRIPTION OF THE DRAWINGS

In the Drawings:

FIG. 1 shows a diagram of a wireless power circuit;

FIG. 2 shows an equivalent circuit;

FIG. 3 shows a diagram of inductive coupling;

FIG. 4 shows a plot of the inductive coupling; and

FIG. 5 shows geometry of an inductive coil.

DETAILED DESCRIPTION

FIG. 1 is a block diagram of an inductively coupled energy transmission system between a source 100, and a load 150. The source includes a power supply 102 with internal impedance Z_(s) 104, a series resistance R₄ 106, a capacitance C1 108 and inductance L1 110. The LC constant of capacitor 108 and inductor 110 causes oscillation at a specified frequency.

The secondary 150 also includes an inductance L2 152 and capacitance C2 154, preferably matched to the capacitance and inductance of the primary. A series resistance R2 156. Output power is produced across terminals 160 and applied to a load ZL 165 to power that load. In this way, the power from the source 102 is coupled to the load 165 through a wireless connection shown as 120. The wireless communication is set by the mutual inductance M.

FIG. 2 shows an equivalent circuit to the transmission system of FIG. 1. The power generator 200 has internal impedance Zs 205, and a series resistance R1 210. Capacitor C1 215 and inductor L1 210 form the LC constant. A current I1 215 flows through the LC combination, which can be visualized as an equivalent source shown as 220, with a value U1.

This source induces into a corresponding equivalent power source 230 in the receiver, to create an induced power U2. The source 230 is in series with inductance L2 240, capacitance C2 242, resistance R2 244, and eventually to the load 165.

Considering these values, the equations for mutual inductance are as follows:

U₂=jωMI₁

U₁=jωMI₂

where:

z_(M) = j ω M $z_{1} = {z_{s} + R_{1} + {j\left( {{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}}} \right)}}$ $z_{2} = {z_{L} + R_{2} + {j\left( {{\omega \; L_{21}} - \frac{1}{\omega \; C_{2}}} \right)}}$ z_(s) = R_(s) + jX_(s) z_(L) = R_(L) + jX_(L)

The Mesh equations are:

U_(s) + U₁ − z₁I₁ = 0 → I₁ = (U_(s) + U₁)/z₁ U₂ − z₂I₂ = 0  I₂ = U₂/z₂ $I_{1} = {{\frac{U_{s} + {z_{M}I_{2}}}{z_{1}}\mspace{14mu} I_{2}} = {\left. \frac{z_{M}I_{1}}{z_{2}}\rightarrow I_{2} \right. = {\frac{z_{M}\left( {U_{s} + {z_{M}I_{2}}} \right)}{z_{1}z_{2}} = {\left. \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}\rightarrow I_{1} \right. = {{\frac{z_{M}}{z_{M}} \cdot I_{2}} = \frac{z_{2}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}}}}}}$

where:

Source power:

P ₁=Re{U _(s) ·I* ₁ }=U _(s)·Re{I* ₁} for avg{U _(s)}=0

Power into load:

P ₂ =I ₂ ·I* ₂Re{z _(L) }=|I ₂|²·Re{z _(L) }=|I ₂|² ·R _(L)

Transfer efficiency:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{I_{2} \cdot I_{2}^{*}}R_{L}}{U_{s}{Re}\left\{ I_{1}^{*} \right\}}}$ ${I_{2} \cdot I_{2}^{*}} = \frac{z_{M}z_{M}^{*}U_{s}^{2}}{\left( {{z_{1}z_{2}} - z_{M^{2}}} \right)\left( {{z_{1}^{*}z_{2}^{*}} - z_{M^{2}}^{*}} \right)}$ ${{Re}\left\{ I_{1}^{*} \right\}} = {{Re}\left\{ \frac{z_{2}^{*}U_{s}}{{z_{1}^{*}z^{*}} - z_{M^{2}}^{*}} \right\}}$

Overall transfer Efficiency is therefore:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{U_{s}^{2} \cdot R_{L}}z_{M}z_{M}^{*}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{*}z_{2}^{*}} - z_{M^{2}}^{*}} \right)}}$ Def.:  z¹ = z₁z₂ − z_(M²) $\left. \begin{matrix} {\left. \rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = {\frac{R_{L}z_{M}z_{M}^{*}}{z^{\prime}z^{*}{Re}\left\{ \frac{z_{2}^{*}z^{\prime}}{z^{\prime}z^{\prime*}} \right\}} = \frac{R_{L}z_{M}z_{M}^{*}}{{Re}\left\{ {z_{2}^{*} \cdot z^{\prime}} \right\}}}}} \\ {= {\frac{R_{L}z_{M}z_{M}^{*}}{{Re}\left\{ {z_{2}^{*}\left( {{z_{1}z_{2}} - {z`}_{M}^{2}} \right)} \right\}} = \frac{\left. R_{L} \middle| z_{M} \right|^{2}}{{Re}\left\{ z_{1} \middle| z_{2} \middle| {}_{2}{{- z_{2}^{*}}z_{M}^{2}} \right\}}}} \end{matrix}\rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = \frac{\left. R_{L} \middle| z_{M} \right|^{2}}{\left| z_{2} \middle| {}_{2}{{{\cdot {Re}}\left\{ z_{1} \right\}} - {z_{M}^{2}{Re}\left\{ z_{2}^{*} \right\}}} \right.}}$ Re{z₁} = R_(s) + R₁ ${{Re}\left\{ z_{2}^{*} \right\}} = {\left. {R_{L} + R_{2}} \middle| z_{2} \right|^{2} = {\left. {\left( {R_{L} + R_{2}} \right)^{2} + \left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right)^{2}} \middle| z_{M} \right|^{2} = {\omega^{2}M^{2}}}}$ z_(M²) = (j ω M)² = −ω²M²

A Transfer efficiency equation can therefore be expressed as:

$\begin{matrix} {\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}{M^{2} \cdot R_{L}}}{{\left( {R_{s} + R_{n}} \right)\left\lbrack {\left( {R_{L} + R_{2}} \right)^{2} + \left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right)} \right\rbrack} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}} \\ \; \end{matrix}$

Which reduces in special cases as follows:

-   A) when ω=ω₀=1/√{square root over (L₂C₂)}, X_(L)=0 or where

${{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + {X_{L}(\omega)}} = 0$ $\begin{matrix} {\eta = {\frac{P_{2}}{P_{1}} = {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {R_{s} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack} \cdot \frac{R_{L}}{\left( {R_{L} + R_{2}} \right)}}}} \\ \; \end{matrix}$

-   B) when ω=ω₀, R_(s)=0:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega_{0}^{2}M^{2}R_{L}}{{R_{1}\left( {R_{L} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$

-   C) when ω=ω₀, R_(s)=0 R_(L)=R₂:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega_{0}^{2}M^{2}}{{4R_{1}R_{2}} + {2\omega_{0}^{2}M^{2}}}}$

-   D) when ω=ω₀, R_(s)=0 R_(L)=R₂ 2R₁R₂>>ω₀ ²M²:

$\eta = {\frac{P_{2}}{P_{1}} \cong {\frac{\omega_{0}^{2}M^{2}}{4R_{1}R_{2}}\mspace{14mu} \left( {{weak}\mspace{14mu} {coupling}} \right)}}$

where: Mutual inductance:

M=k√{square root over (L₁L₂)} where k is the coupling factor

Loaded Q factors:

$Q_{1,L} = {{\frac{\omega \; L_{1}}{R_{s} + R_{1}}\mspace{14mu} Q_{2,L}} = \frac{\omega \; L_{2}}{R_{L} + R_{2}}}$

Therefore, the transfer efficiency in terms of these new definitions:

-   A) when ω=ω₀

$\eta = {\frac{P_{2}}{P_{1}} = {\frac{k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}{1 + {k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}}$ $\begin{matrix} {\eta = {\frac{k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}{1 + {k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}} \\ \; \end{matrix}$

-   C) when ω=ω₀, R_(L)=R₂, (R_(s)=0):

-   D) ω=ω₀, R_(L)=R₂, (R_(s)=0)

2R_(n)R₂>> ω₀²M²− > 1>> k²Q_(1, UL)Q_(2, UL)/2 $\begin{matrix} {\eta = {\frac{P_{2}}{Pn} \cong {\frac{k^{2}Q_{1,{UL}}Q_{2,{UL}}}{4}\mspace{14mu} \left( {{weak}\mspace{14mu} {coupling}} \right)}}} \\ \; \end{matrix}$

Q_(UL): Q unloaded

${Q_{1,{UL}} = \frac{\omega \; L_{1}}{R_{1}}};$ $Q_{2,{UL}} = \frac{\omega \; L_{2}}{R_{2}}$

This shows that the output power is a function of input voltage squared

P₂ = f(U_(s)²) = I₂ ⋅ I₂^(*)R_(L); $I_{2} = \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}$ $P_{2} = {\frac{z_{M}z_{M}^{*}R_{L}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{*}z_{2}^{*}} - z_{M}^{2^{*}}} \right)} \cdot U_{s}^{2}}$ $P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{z_{1}z_{2}z_{1}^{*}z_{2}^{*}} + {z_{M}} + {{z_{M}}^{2} \cdot \left( {{z_{1}z_{2}} + {z_{1}^{*}z_{2}^{*}}} \right)}}$ $P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{{z_{1}z_{2}}}^{2} + {{z_{M}}^{2}2{Re}\left\{ {z_{1}z_{2}} \right\}} + {z_{M}}^{4}}$ z_(M) = jω M z_(M)^(*) = −jω M z_(M) = ω M = z_(M)z_(M)^(*) z_(M)^(*) = −ω²M² = −z_(M)² z_(M)^(2^(*)) = −ω²M² = z_(M)² = −z_(M)² z_(M)² ⋅ z_(M)^(2^(*)) = z_(M)⁴ z₁z₂ = z₁ ⋅ z₂ z₁z₂ + z₁^(*)z₂^(*) = 2Re{z₁z₂} z₁ ⋅ z₂² = z₁² ⋅ z₂²

Definitions:

z₁ = R₁^(′) + jX₁; z₂ = R₂^(′) + jX₂|z₁z₂|² = (R₁^(′2) + X₁²)(R₂^(′2) + X₂²) = R₁^(′2)R₂^(′2) + X₁²R₂^(′2) + X₂²R₁^(′2) + X₁²X₂² Re{z₁z₂} = Re(R₁^(′) + jX₁)(R₂^(′) + jX₂) = R₁^(′)R₂^(′) + X₁X₂|z_(M)| = X_(M) $P_{2} = \frac{X_{M}^{2}{R_{1} \cdot U_{s}^{2}}}{{R_{1}^{\prime 2}R_{2}^{\prime 2}} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{1}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + {2X_{M}^{2}X_{1}X_{2}} + X_{M}^{4}}$ $P_{2} = \frac{X_{M}^{2}{R_{L} \cdot U_{s}^{2}}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{2}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}X_{1}X_{2}}}$

Therefore, when at or near the resonance condition:

ω = ω₀ = ω₂ = ω₀ → X₁ = 0, X₂ = 0 $P_{2} = {\frac{X_{M}^{2}{R_{L} \cdot U_{s}^{2}}}{{R_{j}^{\prime 2}R_{2}^{\prime 2}} + {2X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{4}} = {\frac{X_{M}^{2}R_{L}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2}} \cdot U_{s}^{2}}}$ $P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{{\left( {R_{s} + R_{1}} \right)^{2}\left( {R_{1} + R_{2}} \right)^{2}} + {2\omega_{0}^{2}{M^{2}\left( {R_{s} + R_{1}} \right)}\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{4}M^{4}}} \cdot U_{s}^{2}}$ $\begin{matrix} {P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right)^{2}}U_{s}^{2}}} \\ \; \end{matrix}$

Showing that the power transfer is inversely proportional to several variables, including series resistances.

Mutual inductance in terms of coupling factors and inductions:

$M = {k \cdot \sqrt{L_{1}L_{2}}}$ $\begin{matrix} {P_{2} = {\frac{\omega_{0}^{2}k^{2}L_{1}{L_{2} \cdot R_{L}}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}k^{2}L_{1}L_{2}}} \right)^{2}} \cdot U_{s}^{2}}} \\ {= {\frac{k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}{\left( {1 + {k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}} \right)^{2}} \cdot \frac{U_{s}^{2}R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \end{matrix}$ $\begin{matrix} {P_{2} = {\frac{k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}{\left( {1 + {k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}} \right)^{2}} \cdot \frac{R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} \cdot U_{s}^{2}}} \\ \; \end{matrix}$

The power output is proportional to the square of the input power, as described above. However, there is a maximum input power beyond which no further output power will be produced. These values are explained below. The maximum input power P_(1max) is expressed as:

${P_{1,\max} = {\frac{U_{s}^{2}}{R_{s} + R_{{in},\min}} = {{Re}\left\{ {U_{s} \cdot I_{1}^{*}} \right\}}}};$

R_(in,min): min. permissible input resistance Efficiency relative to maximum input power:

$\eta^{\prime} = {\frac{P_{2}}{P_{1,\max}} = \frac{P_{2}\left( U_{s}^{2} \right)}{P_{1,\max}}}$

Under resonance condition ω=ω₁=ω₂=ω₀:

$\begin{matrix} {\eta^{\prime} = \frac{\omega_{0}^{2}M^{2}{R_{L}\left( {R_{s} + R_{{in},\min}} \right)}}{\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}}} \\ \; \end{matrix}$

Equation for input power (P₁) under the resonance condition is therefore:

$P_{1} = {\frac{P_{2}}{\eta} = {\frac{\omega_{0}^{2}M^{2}{R_{L}\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack}\left( {R_{L} + R_{2}} \right)}{{\left\lbrack {{\left( {R_{s} + R_{2}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{0}M^{2}}} \right\rbrack^{2} \cdot \omega_{0}^{2}}M^{2}R_{L}} \cdot U_{s}^{2}}}$ $\begin{matrix} {P_{1} = {\frac{R_{L} + R_{2}}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} +_{0}^{2}M^{2}} \cdot U_{s}^{2}}} \\ \; \end{matrix}$ For  (R_(s) + R_(M))(R_(L) + R₂)>> ω₀²M²: $P_{1} \cong \frac{U_{S}^{2}}{\left( {R_{s} + R_{1}} \right)}$

The current ratio between input and induced currents can be expressed as

$\frac{I_{2}}{I_{1}} = {\frac{z_{M} \cdot U_{s} \cdot \left( {{z_{1}z_{2}} - z_{n}^{2}} \right)}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)z_{2}U_{s}} = {\frac{z_{M}}{z_{2}} = \frac{{j\omega}\; M}{R_{L} + R_{2} + {j\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}}} \right)}}}}$ ${{at}\mspace{14mu} \omega} = {\omega_{0} = \frac{1}{\sqrt{L_{2}C_{2}}}}$ $\begin{matrix} {\frac{I_{2}}{I_{1}} =} & \frac{{j\omega}\; M}{R_{1} + R_{2}} \\ \; & \; \end{matrix}$ ${{avg}.\left\{ \frac{I_{2}}{I_{1}} \right\}} = \frac{\pi}{2}$

Weak coupling: R₁+R₂>|jωM|

-   →I₂<I₁     Strong coupling: R₁+R₂<|jωM| -   →I₂>I₁     Input current I₁: (under resonance condition)

$I_{1} = {\frac{P_{1}}{U_{s}} = \frac{\left( {R_{1} + R_{2}} \right) \cdot U_{s}}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}}}$ $\begin{matrix} {I_{1} = {\frac{\left( {R_{L} + R_{2}} \right)}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}} \\ \; \end{matrix}$

Output current I₂: (under resonance condition)

$\begin{matrix} {I_{2} = {\frac{{j\omega}\; M}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}} \\ \; \end{matrix}$

Maximizing transfer efficiency and output power (P₂) can be calculated according to the transfer efficiency equation:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}M^{2}R_{L}}{{\left( {R_{s} + R_{n}} \right)\left\lbrack {\left( {R_{L} + R_{2}} \right)^{2} + \underset{}{\left( {{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} \right)^{2}}} \right\rbrack} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$

After reviewing this equation, an embodiment forms circuits that are based on observations about the nature of how to maximize efficiency in such a system.

Conclusion 1)

-   η(L₂, C₂, X_(L)) reaches maximum for

${{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} = 0$

-   That is, efficiency for any L, C, X at the receiver is maximum when     that equation is met.     Transfer efficiency wide resonance condition:

$\eta = {{\frac{P_{2}}{P_{1}}_{\omega = \omega_{0}}} = {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {\underset{}{R_{s}} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack} \cdot \frac{R_{1}}{\left( {R_{L} + R_{2}} \right)}}}$

Conclusion 2)

-   To maximise η R_(S) should be R_(S)<<R₁ -   That is, for maximum efficiency, the source resistance R_(S) needs     to be much lower than the series resistance, e.g., 1/50, or     1/100^(th) or less     Transfer efficiency under resonance and weak coupling condition:

(R_(s) + R_(n))(R_(L) + R₂)>> ω₀²M² $\eta \cong \frac{\omega_{0}^{2}{M^{2} \cdot \overset{}{R_{L}}}}{\left( {R_{s} + R_{n}} \right)\left( {\underset{}{R_{L}} + R_{2}} \right)^{2}}$

Maximising η(R_(L)):

$\frac{\eta}{R_{L}} = {{\frac{\omega_{0}^{2}M^{2}}{R_{s} + R_{1}} \cdot \frac{\left( {R_{L} + R_{2}} \right) - {2R_{L}}}{\left( {R_{L} + R_{2}} \right)^{3}}} = {{0->R_{L}} = R_{2}}}$

Conclusion 3)

-   η reaches maximum for R_(L)=R₂ under weak coupling condition. -   That is, when there is weak coupling, efficiency is maximum when the     resistance of the load matches the series resistance of the     receiver. -   Transfer efficiency under resonance condition. -   Optimizing R_(L) to achieve max. η

$\begin{matrix} {\mspace{79mu} {{{\frac{\eta}{R_{L}} = 0};}\mspace{79mu} {{\frac{}{R_{L}} \cdot \frac{\omega_{0}^{2}M^{2}R_{L}}{{\underset{R_{1}}{\left( \underset{}{R_{s} + R_{1}} \right)^{2}}\left( {R_{L} + R_{2}} \right)^{2}} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}\frac{u}{v}}\mspace{79mu} {\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0}\mspace{79mu} {{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}} = 0}\mspace{79mu} {{u = {\omega_{0}^{2}{M^{2} \cdot R_{L}}}};}\mspace{79mu} {u^{\prime} = {\omega_{0}^{2}M^{2}}}\mspace{79mu} {v = {{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{1} + R_{2}} \right)}}}}\mspace{79mu} {v^{\prime} - {2{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}}}}} & \; \\ \begin{matrix} {{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}} = {{\omega_{0}^{2}M^{2}{R_{L}\left( {{2{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}}} \right)}} -}} \\ {{\left( {{R_{1}^{\prime}\left( {R_{1} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}} \right)\omega_{0}^{2}M^{2}}} \\ {= 0} \\ {= {{2R_{1}^{\prime}{R_{L}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}R_{L}} - {R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}^{2} -}} \\ {{\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}} \\ {= 0} \\ {= {{2R_{1}^{\prime}R_{L}^{2}} + {2R_{1}^{\prime}R_{2}R_{L}} + {\omega_{0}^{2}M^{2}R_{L}} - {R_{1}^{\prime}R_{L}^{2}} - {2R_{1}^{\prime}R_{2}R_{L}} - {R_{1}^{\prime}R_{2}^{2}} -}} \\ {{{\omega_{0}^{2}M^{2}R_{L}} - {\omega_{0}^{2}M^{2}R_{2}}}} \\ {= 0} \\ {= {{\left( {{1R_{1}^{\prime}} - R_{1}^{\prime}} \right)R_{L}^{2}} - {R_{1}^{\prime}R_{2}^{2}} - {\omega_{0}^{2}M^{2}R_{2}}}} \\ {= 0} \end{matrix} & \; \\ {\mspace{79mu} {R_{L}^{2} = \frac{{R_{1}^{\prime}R_{2}^{2}} + {\omega_{0}^{2}M^{2}R_{2}}}{R_{1}^{\prime}}}} & \; \\ {R_{L} = {{\pm \sqrt{\frac{{\left( {R_{s} + R_{1}} \right)R_{2}^{2}} + {\omega_{0}^{2}M^{2}R^{2}}}{\left( {R_{s} + R_{1}} \right)}}} = {{\pm R_{2}} \cdot \sqrt{\frac{\left( {R_{s} + R_{1}} \right) + {\omega_{0}^{2}{M^{2}/R_{2}}}}{\left( {R_{s} + R_{1}} \right)}}}}} & \; \\ {\mspace{79mu} {R_{L,{opt}} = {R_{2}\sqrt{1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}}}}} & \; \end{matrix}$

Weak coupling condition ω₀ ²M²<<(R_(s)+R₁)R₂

R _(L,opt) ≅R ₂

Conclusion 4)

-   There exists an optimum R_(L)>R₂ maximising η     Output power P₂:

$P_{2} = \frac{X_{M}^{2}R_{1}U_{s}^{w}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}\underset{}{X_{2}^{2}}} + {R_{2}^{\prime 2}\underset{}{X_{1}^{2}}} + \underset{}{X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}\underset{}{X_{2}X_{2}}}}$

Conclusion 5)

-   Output power P₂(X₁, X₂) reaches maximum for

$X_{1} = {{{\omega \; L_{1}} - \frac{1}{\omega \; C_{1}} + X_{s}} = 0}$ $X_{2} = {{{\omega \; L_{2}} - \frac{1}{\omega \; C_{2}} + X_{L}} = 0}$

that is, when there is a resonance condition at both the primary and the secondary. Output power P₂ wide resonance condition:

$P_{2} = {\frac{\omega_{0}^{2}{M^{2} \cdot R_{L}}}{\left\lbrack {{\left( {\underset{}{P_{s}} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}} \cdot U_{s}^{2}}$

Conclusion 6)

-   To maximize P₂, R_(S) should be R_(S)<<R₁     Output power P₂ for the wide resonance and weak coupling condition:

(R_(s) + R₁)(R_(L) + R₂)>> ω₀²M² $P_{2} \cong {\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( {R_{s} + R_{1}} \right)^{2}\left( {R_{L} + R_{2}} \right)^{1}} \cdot U_{s}^{2}}$

Conclusion 7)

-   P₂(R_(L)) reaches maximum for R_(L)=R₂ (see conclusion 3) -   For each of the above, the >> or << can represent much greater, much     less, e.g., 20× or 1/20 or less, or 50× or 1/50^(th) or less or 100×     or 1/100^(th) or less.     The value R_(L) can also be optimized to maximize P₂:

$\begin{matrix} {\mspace{79mu} {{\frac{P_{2}}{R_{L}} = 0}\mspace{79mu} {\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0}\mspace{79mu} {{u = {\omega_{0}^{2}M^{2}R_{L}}};}\mspace{79mu} {u^{\prime} = {\omega_{0}^{2}M^{2}}}\mspace{79mu} {v = \left\lbrack {{\left( R_{1}^{\prime} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}}\mspace{79mu} {v^{\prime} = {2 \cdot \left\lbrack {{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack \cdot R_{1}^{\prime}}}{{{\omega_{0}^{2}{M^{2} \cdot R_{L} \cdot {2\left\lbrack {{R_{1}^{\prime}\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack}}R_{1}} - {{\left\lbrack {{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2} \cdot \omega_{0}^{2}}M^{2}}} = 0}{{{2{R_{L}\left( {{R_{1}^{\prime 2}R_{L}} + {R_{1}^{\prime 2}R_{2}}} \right)}} + {1R_{L}\omega_{0}^{2}{M^{2} \cdot R_{1}^{\prime}}} - \left\lbrack {{R_{1}^{\prime}R_{L}} + {R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}} = 0}{{{2R_{1}^{\prime 2}R_{L}^{2}} + {2R_{1}^{\prime 2}R_{2}R_{L}} + {2\omega_{0}^{2}M^{2}R_{1}^{\prime}R_{L}} - {R_{1}^{\prime 2}R_{L}^{2}} - {R_{1}^{\prime 2}R_{2}^{2}} - {\omega_{0}^{2}M^{4}} - {2R_{1}^{\prime 2}R_{2}R_{L}} - {2R_{1}^{\prime}\omega_{0}^{2}M^{2}R_{L}} - {2R_{1}^{\prime}R_{2}\omega_{0}^{2}M^{2}}} = {0 = {{{\left( {{2R_{1}^{\prime 2}} - R_{1}^{\prime 2}} \right)R_{L}^{2}} - {R_{1}^{\prime 2}R_{2}^{2}} - {2R_{1}^{\prime}R_{2}\omega_{0}^{2}M^{2}} - {\omega_{0}^{2}M^{4}}} = {0 = {{{R_{1}^{\prime 2} \cdot R_{L}^{2}} - \left( {{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right)^{2}} = 0}}}}}}} \\ {\mspace{79mu} {R_{L}^{2} = \frac{\left( {{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right)^{2}}{R_{1}^{\prime 2}}}} \\ {\mspace{76mu} {R_{L,{opt}} = {\frac{{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}}{R_{1}^{\prime}} = {R_{2}\left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}}}} \\ {\mspace{79mu} \begin{matrix} {R_{L,{opt}} = {R_{2} \cdot \left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}} & {{Weak}\mspace{14mu} {coupling}\text{:}} \\ \; & {R_{L,{opt}}\overset{\cong}{>}R_{2}} \end{matrix}} \end{matrix}$

Conclusion 8)

-   There exists an optimum R_(L)>R₂ maximizing P₂. This R_(1opt)     differs from the R_(1,opt) maximizing η. -   One embodiment operates by optimizing one or more of these values,     to foam an optimum value.

Inductive coupling is shown with reference to FIGS. 3, 4

FIG. 5 illustrates the Inductance of a multi-turn circular loop coil

$R_{m} = \frac{R_{0} + R_{1}}{2}$ $\begin{matrix} {{Wheeler}\mspace{14mu} {formula}\mspace{14mu} ({empirical})} & \left\lbrack {{Wheeler},{H.\; A.},{``{{Simple}\mspace{14mu} {inductance}}}} \right. \\ {L = \frac{0.8{R_{m}^{2} \cdot N^{2}}}{{6R_{m}} + {9w} + {10\left( \left( {R_{o} + R_{1}} \right) \right.}}} & {{{{{formulas}\mspace{14mu} {for}\mspace{14mu} {radio}\mspace{14mu} {coils}}"}.\; {Proc}.\; I}\; R\; E} \\ \; & \left. {{{Vol}\mspace{14mu} 16},{{{pp}.\mspace{14mu} 1328}\text{-}1400},{{Oct}.\mspace{14mu} 1928.}} \right\rbrack \\ {{Note}\text{:}\mspace{14mu} {this}\mspace{14mu} i\mspace{14mu} {accurate}\mspace{14mu} {if}\mspace{14mu} {all}\mspace{14mu} {three}\mspace{14mu} {terms}\mspace{14mu} {in}} & \; \\ {{denominator}\mspace{14mu} {are}\mspace{14mu} {about}\mspace{14mu} {{equal}.}} & {\lbrack L\rbrack \mu \; H} \\ \; & {\left\lbrack {R_{m},R_{i},R_{0},\omega} \right\rbrack = {inch}} \\ {{{Conversion}\mspace{14mu} {to}\mspace{14mu} H},{m\mspace{14mu} {units}\text{:}}} & {{1\mspace{14mu} m} = {\frac{\overset{\overset{\wp}{}}{10}00}{- 154} \cdot {inch}}} \\ {L = \frac{0.8{R_{m}^{2} \cdot N^{2}}}{{6{R_{m} \cdot \wp}} + {9 \cdot w \cdot \wp} + {10\left( {R_{0} - R_{1}} \right)\wp}}} & \begin{matrix} {{1\mspace{14mu} H} = {10^{6}\mspace{14mu} \mu \; H}} \\ {\lbrack L\rbrack = H} \end{matrix} \\ {L = \frac{0.8 \cdot R_{m}^{2} \cdot \wp^{2} \cdot N^{2} \cdot 10^{- 6}}{{6R_{m}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}}} & {\left\lbrack {R_{m}R_{0}R_{1}\omega} \right\rbrack = m} \end{matrix}$

In standard form:

${L = \frac{\mu_{0} \cdot A_{m} \cdot N^{2}}{K_{c}}};$ A_(m) = π ⋅ R_(m)² μ₀ = 4π ⋅ 10⁻⁷ $L = \frac{0.8 \cdot \wp \cdot 10^{- 6} \cdot {\overset{}{\pi \; R}}_{m}^{2} \cdot N^{2} \cdot \overset{}{4{\pi \cdot 10^{- 7}}}}{{\pi \cdot 4}{\pi \cdot 10^{- 7}}\left( {{6R_{m}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ $L = {{\frac{\mu_{0}{A_{m} \cdot N^{2} \cdot 0.8}{{\wp 10}^{- 6} \cdot 10}}{4{\pi^{2} \cdot 10^{- 7}}\left( {{6R_{m}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}\frac{1}{K_{c}}\mspace{14mu} R_{m}} = \sqrt{\frac{A_{m}}{\pi}}}$ $K_{c} = \frac{{\pi^{2} \cdot 25.4}\left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}{2 \cdot 1000}$ $K_{c} \cong {\frac{1}{8} \cdot \left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ ${L = \frac{\mu_{0}A_{m}N^{2}}{K_{c}}};$ $A_{m} = {{\left( \frac{\left( {R_{0} + R_{1}} \right)}{2} \right)^{2} \cdot {\pi \lbrack L\rbrack}} = H}$

The inductance of a single-turn circular loop is given as:

$K_{c} = \frac{R_{m} \cdot \pi}{\wp \left\lbrack {\frac{8R_{m}}{6} - 2} \right\rbrack}$ ${L = \frac{\mu_{0}A_{m}}{K_{c}}};$ A_(m) = R_(m)² ⋅ π[L] = H

where:

R_(m): mean radius in m

b: wire radius in m,

For a Numerical example:

R₁=0.13 m

R₀=0.14 m

ω=0.01 m

N=36

→L=0.746 mH

The measured inductance

L_(meas)=0.85 mH

The model fraction of Wheeler formula for inductors of similar geometry, e.g, with similar radius and width ratios is:

$K_{c} = {\frac{1}{8}\left( {{5\sqrt{\frac{A_{m}}{\pi}}} + {9w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$ $D = \sqrt{W^{2} + \left( {R_{0} - R_{1}} \right)^{2}}$ $R_{m} = \frac{R_{0} + R_{1}}{2}$

Using a known formula from Goddam, V. R., which is valid for w>(R₀−R₁)

$L = {{0.03193 \cdot R_{m} \cdot N^{2}}\left\lfloor {{2.303\left( {1 + \frac{w^{2}}{32R_{m}^{2}} + \frac{D^{2}}{96R_{m}^{2}}} \right){\log \left( \frac{8R_{m}}{D} \right)}} - \wp + \frac{w_{1/2}^{2}}{16R_{m}^{2}}} \right\rfloor}$

1w H,m units:

$L = {\mu_{0}{R_{m} \cdot {N^{2}\left\lbrack {{\left( {1 + \frac{w^{2}}{32R_{m}^{2}} + \frac{D^{2}}{96R_{m}^{2}}} \right){\ln \left( \frac{8R_{m}}{D} \right)}} - \wp + \frac{w^{2}y_{2}}{16R_{m}^{2}}} \right\rbrack}}}$

EXAMPLE 1

R₁ = 0.13  m R₀ = 0.14  m W = 0.01  m N = 36 L = 757  μH ${{Ratio}\text{:}\mspace{14mu} \frac{W}{R_{0} - R_{1}}} = {{1->y_{1}} = 0.8483}$ y₂ = 0.816

From [Terman, F.] EXAMPLE 2 Given in [Goddam, V. R.]

R₀ = 8.175  inches R₁ = 7.875  inches W = 2  inches N = 57 y₁ = 0.6310 y₂ = 0.142− > L = 2.5  mH  (2.36  mH) ${{{Ratio}\text{:}\mspace{14mu} \frac{2}{R_{0} - R_{1}}} = {\frac{2}{0.3} = 6.667}}\mspace{14mu}$ or $\frac{R_{0} - R_{1}}{W} = {\frac{0.3}{2} = 0.15}$

where Goddam, V. R. is the Thesis Masters Louisiana State University, 2005, and Terman, F. is the Radio Engineers Handbook, McGraw Hill, 1943.

Any of these values can be used to optimize wireless power transfer between a source and receiver.

From the above, it can be seen that there are really two different features to consider and optimize in wireless transfer circuits. A first feature relates to the way in which efficiency of power transfer is optimized. A second feature relates to maximizing the received amount of power—independent of the efficiency.

One embodiment, determines both maximum efficiency, and maximum received power, and determines which one to use, and/or how to balance between the two.

In one embodiment, rules are set. For example, the rules may specify:

Rule 1—Maximize efficiency, unless power transfer will be less than 1 watt. If so, increase power transfer at cost of less efficiency.

Rule 2—Maximize power transfer, unless efficiency becomes less than 30%.

Any of these rules may be used as design rules, or as rules to vary parameters of the circuit during its operation. In one embodiment, the circuit values are adaptively changes based on operational parameters. This may use variable components, such as variable resistors, capacitors, inductors, and/or FPGAs for variation in circuit values.

Although only a few embodiments have been disclosed in detail above, other embodiments are possible and the inventors intend these to be encompassed within this specification. The specification describes specific examples to accomplish a more general goal that may be accomplished in another way. This disclosure is intended to be exemplary, and the claims are intended to cover any modification or alternative which might be predictable to a person having ordinary skill in the art. For example, other sizes, materials and connections can be used. Other structures can be used to receive the magnetic field. In general, an electric field can be used in place of the magnetic field, as the primary coupling mechanism. Other kinds of antennas can be used. Also, the inventors intend that only those claims which use the-words “means for” are intended to be interpreted under 35 USC 112, sixth paragraph. Moreover, no limitations from the specification are intended to be read into any claims, unless those limitations are expressly included in the claims.

Where a specific numerical value is mentioned herein, it should be considered that the value may be increased or decreased by 20%, while still staying within the teachings of the present application, unless some different range is specifically mentioned. Where a specified logical sense is used, the opposite logical sense is also intended to be encompassed. 

1-18. (canceled)
 19. An apparatus configured to receive wireless power from a transmitter, the apparatus comprising: an inductor having an inductance value; a capacitor electrically connected to the inductor and having a capacitance value; and an optimizing circuit configured to: optimize transfer efficiency of power received wirelessly from the transmitter, provided that an amount of the power received wirelessly and provided to a load is greater than or equal to a received power threshold, or optimize the amount of the power received wirelessly from the transmitter, provided that the power transfer efficiency is greater than or equal to an efficiency threshold.
 20. The apparatus of claim 19, wherein the optimizing circuit is configured to optimize power transfer efficiency or optimize the amount of power received based at least in part on maintaining a first resonant frequency of the transmitter substantially equal to a second resonant frequency of a receive circuit comprising the inductor and capacitor.
 21. The apparatus of claim 19, wherein the optimizing circuit is configured to optimize power transfer efficiency or optimize the amount of power received based at least in part on whether the transmitter is weakly coupled to the receive circuit as compared to when the transmitter is strongly coupled to the receive circuit.
 22. The apparatus of claim 19, wherein the optimizing circuit is configured to maintain a resistance of the inductor substantially equal to a series resistance.
 23. The apparatus of claim 19, wherein the optimizing circuit comprises at least one of a first component configured to vary the inductance value of the inductor, a second component configured to vary the capacitance value of the capacitor, a variable resistor, or an FPGA.
 24. A method for wirelessly receiving power from a transmitter, the, the method comprising: optimizing transfer efficiency of power received wirelessly from the transmitter, provided that an amount of the power received wirelessly and provided to a load is greater than or equal to a received power threshold; or optimizing the amount of the power received wirelessly from the transmitter, provided that the power transfer efficiency is greater or equal to an efficiency threshold.
 25. The method of claim 24, wherein optimizing power transfer efficiency and optimizing the amount of the received power comprises optimizing based on at least one of maintaining a first resonant frequency of the transmitter substantially equal to a second resonant frequency of a receive circuit comprising an inductor and capacitor or based on whether the transmitter is weakly coupled to the receive circuit as compared to when the transmitter is strongly coupled to the receive circuit.
 26. An apparatus configured to receive wireless power from a transmitter, the apparatus comprising: means for optimizing transfer efficiency of power received wirelessly from a transmitter, provided that an amount of the power received wirelessly and provided to a load is greater than or equal to a received power threshold; or means for optimizing the amount of the power received wirelessly, provided that the power transfer efficiency is greater or equal to an efficiency threshold.
 27. The apparatus of claim 26, wherein the means for optimizing power transfer efficiency and optimizing the amount of the received power comprises means for optimizing based on at least one of maintaining a first resonant frequency of the transmitter substantially equal to a second resonant frequency of a receive circuit comprising an inductor and capacitor or based on whether the transmitter is weakly coupled to the receive circuit as compared to when the transmitter is strongly coupled to the receive circuit.
 28. An apparatus configured to transmit wireless power to a receiver, the apparatus comprising: an inductor having an inductance value; a capacitor electrically connected to the inductor and having a capacitance value; and an optimizing circuit configured to: optimize transfer efficiency of power transmitted wirelessly to the receiver, provided that an amount of power received wirelessly and provided to a load of the receiver is greater than or equal to a received power threshold; or optimize the amount of the power received wirelessly, provided that the power transfer efficiency is greater than or equal to an efficiency threshold.
 29. The apparatus of claim 28, wherein the optimizing circuit comprises at least one of a first component configured to vary the inductance value of the inductor, a second component configured to vary the capacitance value of the capacitor, a variable resistor, or an FPGA.
 30. An apparatus for receiving power via a wireless field and for delivering power to a load, the apparatus comprising: a first antenna circuit coupled to the load and comprising an antenna, the first antenna circuit being configured to receive power via the wireless field for powering the load, the received power corresponding to a time-varying voltage signal from a second antenna circuit, the load being characterized by a reactance X and the first antenna circuit being characterized by an inductance L and a capacitance C, at least one of the inductance L and capacitance C configured to maintain the reactance X to be substantially equal to a first value that is inversely proportional to the capacitance C minus a second value that is directly proportional to the inductance L; and a controller coupled to the first antenna circuit and configured to adjust at least one parameter of the first antenna circuit.
 31. The apparatus of claim 30, wherein the first antenna circuit comprises a variable inductor and a variable capacitor, wherein the controller is configured to adjust the inductance L by adjusting the variable inductor, and wherein the controller is configured to adjust the capacitance C by adjusting the variable capacitor.
 32. The apparatus of claim 30, wherein the controller comprises an FPGA.
 33. The apparatus of claim 30, wherein the first antenna circuit comprises a first inductive coil.
 34. The apparatus of claim 30, wherein the received power is coupled from a near-field of the second antenna circuit.
 35. The apparatus of claim 30, wherein the first antenna circuit is configured to oscillate at a frequency substantially equal to a resonant frequency of the first antenna circuit in response to the wireless field produced by the second antenna circuit.
 36. A method of receiving power via a wireless field and for delivering power to a load, the apparatus comprising: wirelessly receiving power at a first antenna circuit for powering the load, the received power corresponding to a time-varying voltage signal from a second antenna circuit, the load being characterized by a reactance X and the first antenna circuit being characterized by an inductance L and a capacitance C, at least one of the inductance L and capacitance C configured to maintain the reactance X to be substantially equal to a first value that is inversely proportional to the capacitance C minus a second value that is directly proportional to the inductance L; and adjusting at least one parameter of the first antenna circuit.
 37. The method of claim 36, wherein the first antenna circuit comprises a variable inductor and a variable capacitor, wherein adjusting the at least one parameter comprises adjusting the variable inductor, and wherein adjusting the at least one parameter comprises adjusting the variable capacitor.
 38. The method of claim 36, wherein wirelessly receiving power comprising coupling power from a near-field of the second antenna circuit.
 39. The method of claim 36, wherein the first antenna circuit is configured to oscillate at a frequency sustainably equal to a resonant frequency of the first antenna circuit in response to the wireless field produced by the second antenna circuit.
 40. An apparatus for receiving power via a wireless field and for delivering power to a load, the apparatus comprising: means for wirelessly receiving power for powering the load, the received power corresponding to a time-varying voltage signal from a second antenna circuit, the load being characterized by a reactance X and the means for wirelessly receiving power being characterized by an inductance L and a capacitance C, at least one of the inductance L and capacitance C configured to maintain the reactance X to be substantially equal to a first value that is inversely proportional to the capacitance C minus a second value that is directly proportional to the inductance L; and means for adjusting at least one parameter of the means for wirelessly receiving power.
 41. The apparatus of claim 40, wherein the means for wirelessly receiving power comprises a first antenna circuit comprising a variable inductor and a variable capacitor, wherein the means for adjusting comprises means for adjusting the variable inductor, and wherein the means for adjusting comprises means for adjusting the variable capacitor.
 42. The apparatus of claim 40, wherein the means for adjusting comprises a controller.
 43. The apparatus of claim 40, wherein the received power is coupled from a near-field of the second antenna circuit.
 44. The apparatus of claim 40, wherein the means for wirelessly receiving power is configured to oscillate at a frequency substantially equal to a resonant frequency of the means for wirelessly receiving power in response to the wireless field produced by the second antenna circuit.
 45. An apparatus for delivering power to a load via a wireless field, the apparatus comprising: a power source configured to output a time-varying voltage signal at a voltage level and characterized by a reactance X; and a first antenna circuit configured to receive the voltage signal from the power source and to output power to a second antenna circuit to power the load via the wireless field, the first antenna circuit characterized by an inductance L and a capacitance C, at least one of the inductance L and capacitance C configured to maintain the reactance X to be substantially equal to a first value that is inversely proportional to the capacitance C minus a second value that is directly proportional to the inductance L. 